Noisy edges on my animation
Don't use branched path tracing.

If you really REALLY REALLY want to use branched path tracing, then you need to increase your Anti-Aliasing samples. For each ONE pass on one pixel, you're doing 1024 Diffuse samples, and 2048 Glossy samples, but only 1 AA sample. Your Diffuse and Glossy settings are multiplied by the number of AA samples you use. Secondly, your AA settings define how many "passes" each pixel is given. You're only making two passes.

...and don't use branched path tracing.

Editorial: The only reason I say not to use branched path tracing (BPT) is that it's a specialized tool for solving a particular problem. Using it for a situation like this is unnecessarily complicated and creates more problems than it solves. It's common for Blenderheads to use BPT because they think of it like a one-button-click that "makes things better," but there's only one thing that BPT can do better: render faster... sometimes.

Don't worry about speed until speed becomes the bottleneck that you need to solve right now. And BPT is only faster if you know how to set it up for your particular use-case. Otherwise, it's likely to create more problems than it solves.

It's like the old adage about optimizing code "The First Rule of Program Optimization: Don't do it. The Second Rule of Program Optimization (for experts only!): Don't do it yet." — Michael A. Jackson

Similarly, the first rule of BPT should be "Don't use it," and the second rule (for experts only!) "Don't use it yet."